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Theorem cvmliftmo 31598
Description: A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by NM, 17-Jun-2017.)
Hypotheses
Ref Expression
cvmliftmo.b 𝐵 = 𝐶
cvmliftmo.y 𝑌 = 𝐾
cvmliftmo.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftmo.k (𝜑𝐾 ∈ Conn)
cvmliftmo.l (𝜑𝐾 ∈ 𝑛-Locally Conn)
cvmliftmo.o (𝜑𝑂𝑌)
cvmliftmo.g (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
cvmliftmo.p (𝜑𝑃𝐵)
cvmliftmo.e (𝜑 → (𝐹𝑃) = (𝐺𝑂))
Assertion
Ref Expression
cvmliftmo (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Distinct variable groups:   𝐶,𝑓   𝑓,𝐺   𝑓,𝐾   𝑓,𝑂   𝜑,𝑓   𝑓,𝐹   𝑃,𝑓
Allowed substitution hints:   𝐵(𝑓)   𝐽(𝑓)   𝑌(𝑓)

Proof of Theorem cvmliftmo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 cvmliftmo.b . . . . 5 𝐵 = 𝐶
2 cvmliftmo.y . . . . 5 𝑌 = 𝐾
3 cvmliftmo.f . . . . . 6 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
43ad2antrr 697 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
5 cvmliftmo.k . . . . . 6 (𝜑𝐾 ∈ Conn)
65ad2antrr 697 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝐾 ∈ Conn)
7 cvmliftmo.l . . . . . 6 (𝜑𝐾 ∈ 𝑛-Locally Conn)
87ad2antrr 697 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝐾 ∈ 𝑛-Locally Conn)
9 cvmliftmo.o . . . . . 6 (𝜑𝑂𝑌)
109ad2antrr 697 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑂𝑌)
11 simplrl 754 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 ∈ (𝐾 Cn 𝐶))
12 simplrr 755 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑔 ∈ (𝐾 Cn 𝐶))
13 simprll 756 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = 𝐺)
14 simprrl 758 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑔) = 𝐺)
1513, 14eqtr4d 2807 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = (𝐹𝑔))
16 simprlr 757 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑓𝑂) = 𝑃)
17 simprrr 759 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑔𝑂) = 𝑃)
1816, 17eqtr4d 2807 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑓𝑂) = (𝑔𝑂))
191, 2, 4, 6, 8, 10, 11, 12, 15, 18cvmliftmoi 31597 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 = 𝑔)
2019ex 397 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) → ((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
2120ralrimivva 3119 . 2 (𝜑 → ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
22 coeq2 5419 . . . . 5 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
2322eqeq1d 2772 . . . 4 (𝑓 = 𝑔 → ((𝐹𝑓) = 𝐺 ↔ (𝐹𝑔) = 𝐺))
24 fveq1 6331 . . . . 5 (𝑓 = 𝑔 → (𝑓𝑂) = (𝑔𝑂))
2524eqeq1d 2772 . . . 4 (𝑓 = 𝑔 → ((𝑓𝑂) = 𝑃 ↔ (𝑔𝑂) = 𝑃))
2623, 25anbi12d 608 . . 3 (𝑓 = 𝑔 → (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)))
2726rmo4 3549 . 2 (∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
2821, 27sylibr 224 1 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  ∃*wrmo 3063   cuni 4572  ccom 5253  cfv 6031  (class class class)co 6792   Cn ccn 21248  Conncconn 21434  𝑛-Locally cnlly 21488   CovMap ccvm 31569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-fin 8112  df-fi 8472  df-rest 16290  df-topgen 16311  df-top 20918  df-topon 20935  df-bases 20970  df-cld 21043  df-nei 21122  df-cn 21251  df-conn 21435  df-nlly 21490  df-hmeo 21778  df-cvm 31570
This theorem is referenced by:  cvmliftlem14  31611  cvmlift2lem13  31629  cvmlift3  31642
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